If `f(x)=4x-5 , g(x)=x^2 and h(x)=1/x` then `f(g(h(x)))` is :

Options

A. `4/(x-5)`
B. `1/(4x-5)^2`
C. `1/(x/4-5)`
D. `(4/(x^2)-5)`

Answer

To solve the problem of finding \( f(g(h(x))) \) given the functions \( f(x) = 4x - 5 \), \( g(x) = x^2 \), and \( h(x) = \frac{1}{x} \), we will follow these steps:

### Step 1: Find \( h(x) \)
First, we need to evaluate \( h(x) \):
\[ h(x) = \frac{1}{x} \]

### Step 2: Substitute \( h(x) \) into \( g(x) \)
Next, we substitute \( h(x) \) into \( g(x) \):
\[ g(h(x)) = g\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^2 = \frac{1}{x^2} \]

### Step 3: Substitute \( g(h(x)) \) into \( f(x) \)
Now, we substitute \( g(h(x)) \) into \( f(x) \):
\[ f(g(h(x))) = f\left(\frac{1}{x^2}\right) = 4\left(\frac{1}{x^2}\right) - 5 \]

### Step 4: Simplify the expression
Now we simplify the expression:
\[ f\left(\frac{1}{x^2}\right) = \frac{4}{x^2} - 5 \]

### Final Result
Thus, the final expression for \( f(g(h(x))) \) is:
\[ f(g(h(x))) = \frac{4}{x^2} - 5 \]